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REVIEW 2 major objections 6 minor 66 references

Tensor resonances contribute only a small fraction of the measured B → PPℓ⁺ℓ⁻ and B → PVℓ⁺ℓ⁻ rates, so other resonances likely dominate.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-31 18:23 UTC pith:6SGY2KSD

load-bearing objection Solid SU(3)-normalized catalog showing tensor pieces sit 1–3 decades below measured B→PP/PVℓℓ totals; the headline inequality survives the usual single-anchor and breaking caveats. the 2 major comments →

arxiv 2607.24315 v1 pith:6SGY2KSD submitted 2026-07-27 hep-ph

Studying the tensor resonance contributions in B to PPell^+ell^- and B to PVell^+ell^- decays

Ru-Min Wang , Xiu-Ping Fan , Si-Yu Xu , Yi Qiao , Xiao-Dong Cheng , Yuan-Guo Xu This is my paper
classification hep-ph
keywords B decaystensor mesonsSU(3) flavor symmetryrare semileptonic decaysfinite width effectsbranching ratiosb → sℓ⁺ℓ⁻b → dℓ⁺ℓ⁻
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper estimates how much light tensor mesons contribute when a B meson decays to two light mesons plus a lepton pair. Using flavor SU(3) to relate all B → Tℓ⁺ℓ⁻ amplitudes and normalizing to the single measured mode Bs → f′₂(1525)μ⁺μ⁻, it predicts the tensor pieces of many four-body rates both in the narrow-width limit and with finite-width smearing. The resulting tensor branching fractions sit well below the total rates already measured for several channels. A sympathetic reader cares because four-body rare B decays are important backgrounds and probes of new physics; knowing that tensors are subdominant tells experimentalists which other resonances (scalars, vectors, axial vectors) to prioritize in amplitude analyses.

Core claim

Once all B → Tℓ⁺ℓ⁻ rates are fixed from the measured Bs → f′₂(1525)μ⁺μ⁻ branching fraction via SU(3) amplitude relations, the tensor-resonance contributions to B → PPℓ⁺ℓ⁻ and B → PVℓ⁺ℓ⁻ are systematically smaller than the corresponding total experimental branching fractions. Finite-width corrections usually reduce the rates modestly, open small near-threshold and subthreshold channels, and do not reverse the conclusion that tensors are subdominant.

What carries the argument

SU(3) flavor relations among the hadronic amplitudes of B → Tℓ⁺ℓ⁻ (retaining only the leading nonperturbative coefficient a0), normalized to the single experimental input B(Bs → f′₂(1525)μ⁺μ⁻) and then folded with T → PP/PV branching fractions under narrow-width and finite-width treatments.

Load-bearing premise

All rates are computed in the exact SU(3) limit with only one overall strength parameter; the paper drops the breaking and annihilation terms because existing data cannot fix them.

What would settle it

A partial-wave or angular analysis of Bs → KKℓ⁺ℓ⁻ or B → Kπℓ⁺ℓ⁻ that isolates a tensor fraction comparable to the total measured rate (rather than the few-percent level predicted here) would falsify the claim that tensors are subdominant.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Tensor components can be treated as small, known inputs in future amplitude analyses of B → PPℓ⁺ℓ⁻ and B → PVℓ⁺ℓ⁻.
  • Scalar, vector, or axial-vector resonances (or their excitations) should account for most of the measured rates in channels such as Bs → π⁺π⁻μ⁺μ⁻ and B⁺ → ϕK⁺μ⁺μ⁻.
  • Bs → KKℓ⁺ℓ⁻ modes, where the tensor pieces are relatively larger, are the most promising places to hunt for spin-2 angular signatures.
  • Finite-width tails can open otherwise forbidden subthreshold modes such as K₂*(1430) → Kη′, offering a qualitative handle on resonance line shapes.
  • The same SU(3)-normalized tensor rates supply first estimates for previously unquoted modes such as Bs → f₂(1270)ℓ⁺ℓ⁻ and Bd → f′₂(1525)ℓ⁺ℓ⁻.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If tensors remain subdominant after more precise form-factor and SU(3)-breaking studies, experimental programs can safely de-emphasize pure tensor searches and focus resources on scalar and axial-vector partial waves.
  • The large gap between predicted tensor rates and existing upper limits on B → MMτ⁺τ⁻ suggests those channels will first constrain other resonances or non-resonant continuum rather than tensors.
  • Once energy-dependent total widths for K₂* and f′₂ are available, the near-threshold finite-width enhancements quoted here can be turned into quantitative predictions and used as line-shape diagnostics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The manuscript studies semileptonic B → Tℓ⁺ℓ⁻ decays (T = a₂(1320), K*₂(1430), f₂(1270), f′₂(1525)) and the cascade four-body modes B → T(→PP,PV)ℓ⁺ℓ⁻ using flavor SU(3). The effective Hamiltonian, B→T form-factor decomposition (Eq. 4), angular observables (Eqs. 6–10), and SU(3) amplitude expansion (Eq. 17, Tab. I) are standard. Since only B(B⁰s → f′₂(1525)μ⁺μ⁻) = (1.62 ± 0.22)×10⁻⁷ is measured, all rates are normalized to this single input in three schemes: S1 (q²-independent effective hadronic factor H_{B→T} = 24.70 ± 2.34), SPQCD and SLCSR (V(0) extracted from the anchor mode, form-factor ratios rᵢ from external calculations). Cascade rates are obtained via the narrow-width approximation (Eq. 19) and a finite-width Breit–Wigner integral (Eq. 20). The central result is that tensor-resonance contributions to the measured B → PPℓ⁺ℓ⁻ and B → PVℓ⁺ℓ⁻ modes are one to three orders of magnitude below the measured totals, implying that scalar, vector, or axial-vector resonances dominate; finite-width effects are modest except near threshold, and the subthreshold K*₂ → Kη′ channel is (correctly) not quoted numerically because a fixed total width would be unphysical there.

Significance. If the numbers hold, the paper provides a useful, comprehensive map of tensor-resonance contributions to rare four-body B decays at a moment when LHCb is beginning amplitude analyses of these final states. Strengths worth naming: (i) the analysis is explicit about its logical structure — a single-anchor SU(3) estimate rather than an ab initio prediction — and the authors themselves state (§II.C.4) that the three schemes are "phenomenological scenarios rather than three independent SU(3) predictions" and that quoted errors exclude SU(3) breaking; (ii) the subthreshold K*₂(1430) → Kη′ modes are responsibly withheld rather than quoted with an unphysical fixed-width treatment, and near-threshold results are flagged with ♯; (iii) long-distance pollution of the anchor mode and neglected interference (Tab. XV) are disclosed. The central inequality (tensor ≪ measured totals) carries margins of factors ~15 to ~10³ (e.g., B(B⁰s → f′₂(→π⁺π⁻)μμ) ≈ 9.2×10⁻¹⁰ vs. 8.4×10⁻⁸ measured; B(B⁺ → K*₂⁺(→φK⁺)μμ) ≈ 7.6×10⁻¹¹ vs. 7.9×10⁻⁸), so it is robust against the single-anchor normalization and plausible SU(3) breaking. The predictions are falsifiable with forthcoming LHCb/Belle II data, which gives th

major comments (2)
  1. [§II.B, §II.C.4, Tabs. II–XIV] All quoted 1σ errors propagate only the anchor-mode measurement, CKM, lifetimes, masses, and (in SPQCD/SLCSR) the rᵢ ratios; the SU(3)-breaking coefficients a₁, a₂, b₀ of Eq. (17) are set to zero and contribute no uncertainty. The text acknowledges this, but the tables — which are the paper's main deliverable and are presented as predictions to be tested — print errors that a reader will naturally read as total. Since the authors themselves estimate 20–30% amplitude-level breaking (§II.B), i.e. up to ~50–70% at rate level, each table caption (or a header note) should state explicitly that errors are within-scheme only, and ideally one representative table should show the effect of an a₁/a₂ variation of ±25% relative to a₀ to demonstrate that the headline inequalities in §III survive. This does not threaten the central claim (the margins are 1.5–3 decades), but the precision of individual
  2. [§III.A, Refs. [61, 62], Tabs. IX–X, XIII–XIV] The B → T(→PV)ℓ⁺ℓ⁻ tables depend on B(T → PV) inputs attributed to Refs. [61, 62], of which [62] is listed as 'in preparation' (and its title refers to B → PPℓν, which does not obviously match its use here). For reproducibility, the T → PV branching fractions or coupling inputs actually used should be tabulated in an appendix, or the citation clarified; as it stands a reader cannot independently verify Tabs. IX–X and XIII–XIV. This is fixable within the manuscript's scope and should be addressed before publication.
minor comments (6)
  1. [Tab. IV] In the B⁰d → f′₂(1525)e⁺e⁻ and μ⁺μ⁻ rows, the [14.18, q²max] bin entries read (9.32 ± 9.32)×10⁻⁷ and (9.34 ± 9.34)×10⁻⁷ against a row unit of ×10⁻¹¹; these are inconsistent with the quoted totals (9.56 ± 4.06 and 7.86 ± 2.93) and with the corresponding SPQCD entries in Tab. III ((2.07 ± 2.07)×10⁻¹⁰). Almost certainly a units/typo error; please check.
  2. [Tab. VIII] The τ entry for B⁰d → a⁰₂(→π⁰η)ℓ⁺ℓ⁻, (8.94 ± 0.39)×10⁻³, has a much smaller fractional error than its e and μ counterparts (±26% and ±20%) and than neighboring τ entries; please verify the error propagation for this cell.
  3. [§III.B, Eq. (20)] The truncated Breit–Wigner window n = 2 ((m_R ± nΓ_R)²) is a convention; a sentence quantifying the sensitivity of the finite-width results to n (e.g., n = 3) would strengthen the near-threshold discussion, particularly for the f′₂(1525) → ηη′ enhancement.
  4. [§II.B, Eqs. (13), (16)] θ_{f₂} ∈ [8°, 10°] and θ_P ∈ [−20°, −10°] are quoted as ranges, but the text does not state whether the table entries correspond to central values or whether the range is included in the errors. Please clarify; the f₂/f′₂ rates depend sensitively on sin θ_{f₂}.
  5. [Throughout] Minor typographical issues: 'conpletion' (§III.A), 'should be regards as' (captions of Tabs. XI–XIV), 'SQCD' for 'SPQCD' (§II.C.4), and inconsistent spacing in Eq. (3). The definition of the ♯ symbol is repeated verbatim in four table captions; once in the text would suffice.
  6. [§I, Ref. [64]] The post-completion LHCb ττ search [64] is cited with upper limits of 10⁻⁶–10⁻⁴; it would help the reader to state explicitly which of the binned limits overlap the tensor-resonance mass windows, since the comparison as written is to total-rate limits.

Circularity Check

3 steps flagged

Single-anchor SU(3) fit makes the full B→Tℓ⁺ℓ⁻ suite a rescaling of one measured BR; four-body rates further multiply by same-group T→MM inputs—but the headline ‘tensor ≪ measured totals’ comparison is to independent external data and is not forced by construction.

specific steps
  1. fitted input called prediction [§II.C.1 (S1 scheme), after Eq. (18); Table II]
    "After considering the bounds of B(B0_s → f ′_2(1525)µ+µ−) given in Eq. (1), HB→T can be determined, and we obtain HB→T = 24.70 ± 2.34. The estimates of the branching ratios are obtained and listed in the second column of Tab. II."

    HB→T is fitted directly to the sole measured B→T rate. With mℓ² terms dropped, every other e/μ branching ratio in S1 is N_T²λ H_B→T² times SU(3)/kinematic factors, so the sibling ‘predictions’ are rescalings of that one fitted number under the a0-only truncation—not independent dynamical forecasts.

  2. fitted input called prediction [§II.C.2–3 (SPQCD and SLCSR); Table II]
    "We obtain V(0) = 0.22 ± 0.11, which is similar to the PQCD result V(0) = 0.20 ± 0.04 with larger error. The branching ratios of the B → T ℓ+ℓ− decays are predicted by the obtained V(0) = 0.22 ± 0.11... The extracted value of V(0) = 0.13 ± 0.05 is consistent with the LCSR calculation result V(0) = 0.15 ± 0.02 within uncertainties... B(B0_s → f ′_2(1525)µ+µ−) 1.62 ± 0.22 [all three schemes]"

    V(0) is extracted so that the anchor mode reproduces the experimental BR; Fi(0)=ri V(0) with ri from external models. All tabulated B→T rates (and thus all downstream four-body tensor pieces) scale as |V(0)|² fixed by that one datum. The anchor column is reproduced by construction in every scheme.

  3. self citation load bearing [§III.A; Refs. [61,62]; Tabs. VII–XIV]
    "Using the expressions of B(B → T ℓ+ℓ−) in SPQCD scheme, B(T → PP, PV) given in Refs. [61, 62], and relevant experimental data given in PDG [1], we can give the branching ratios of the B → T(→ PP)ℓ+ℓ− and B → T(→ PV)ℓ+ℓ− decays... [62] Yuan-Guo Xu and Ru-Min Wang, B → PP ℓ+νℓ, in preparation."

    Numerical four-body tensor components use B(T→PP/PV) from the same collaboration’s prior paper and an in-preparation companion. Those inputs are load-bearing for the absolute four-body rates that are then declared ‘small,’ though many channels also lean on PDG and the smallness claim is still checked against external totals.

full rationale

The paper’s B→Tℓ⁺ℓ⁻ estimates are obtained by fixing one overall nonperturbative strength (HB→T in S1, or V(0) in SPQCD/SLCSR) to the single measured mode B(B0s→f′2(1525)μ⁺μ⁻) and propagating it through SU(3) amplitude coefficients (and external form-factor ratios ri). Sibling B→T rates are therefore proportional to that anchor by construction under the stated symmetry truncation (a1=a2=b0=0). Four-body tensor-resonance components then multiply by B(T→PP/PV) taken in part from the authors’ prior/in-preparation works. That is a standard normalize-and-extrapolate phenomenology, not a closed logical loop: the load-bearing claim that those tensor components are small relative to measured B→PPℓ⁺ℓ⁻ and B→PVℓ⁺ℓ⁻ totals compares the extrapolated pieces to independent LHCb/PDG totals, with margins of ~15–10³ that the fit structure does not force. Reproducing the anchor itself is tautological and is not sold as a prediction. No uniqueness theorem or ansatz is smuggled in via self-citation to forbid alternatives. Score 4 reflects real fitted-input structure and some same-group T→MM dependence without the central inequality reducing to its inputs.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central numerical claims rest on one experimental normalization, exact SU(3) with a single reduced amplitude, external B → T form-factor shapes, and prior T → PP/PV branching fractions. No new dynamical entity is postulated; the ledger is mostly symmetry assumptions plus fitted overall scales.

free parameters (4)
  • a0 / HB→T (S1) or V(0) (SPQCD, SLCSR) = HB→T = 24.70 ± 2.34; V(0)_PQCD = 0.22 ± 0.11; V(0)_LCSR = 0.13 ± 0.05
    Single overall nonperturbative scale fixed to B(Bs → f′₂(1525)μ⁺μ⁻); all other B → T rates scale with it.
  • form-factor ratios ri = Fi(0)/V(0) = from Refs. [37] and [13], not refit here
    Taken from external PQCD and LCSR calculations and held fixed while only V(0) is renormalized to data; drive SPQCD/SLCSR uncertainties.
  • tensor mixing angle θ_f2 = θ_f2 ∈ [8°, 10°]
    Enters f2–f′2 composition in the SU(3) amplitude table; scanned in a small PDG-motivated range.
  • pseudoscalar mixing angle θ_P = θ_P ∈ [−20°, −10°]
    Used for η–η′ content in T → PP channels.
axioms (5)
  • domain assumption Exact SU(3) flavor symmetry for B → T hadronic amplitudes with only a0 retained; a1, a2 breaking and OZI-suppressed b0 neglected
    Stated explicitly in §II.B after Eq. (17); authors note data cannot fix breaking and 20–30% effects would enlarge errors.
  • domain assumption Standard Model b → q′ℓ⁺ℓ⁻ effective Hamiltonian with Wilson coefficients C7,9,10 from Buchalla et al.
    Eq. (3) and Ref. [49]; no new physics operators.
  • domain assumption Narrow-width factorization B(B → T(→ M1M2)ℓℓ) = B(B → Tℓℓ) B(T → M1M2), and the finite-width integral Eq. (20) with fixed Γ_R and n=2
    §III.A–B; fixed-width choice is flagged as only qualitative for near/subthreshold modes.
  • domain assumption B → T form factors share the same SU(3) flavor coefficients mode-by-mode while q² shapes come from PQCD or LCSR
    §II.A–C; relative Fi(q²) are external dynamical inputs, not derived here.
  • domain assumption T → PP and T → PV branching fractions from the authors’ prior SU(3)/phenomenology papers and PDG
    Refs. [61,62] and PDG [1]; feed all four-body numbers.

pith-pipeline@v1.2.0-grok45-kimik3 · 41898 in / 3636 out tokens · 82339 ms · 2026-07-31T18:23:29.897123+00:00 · methodology

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read the original abstract

We analyze the semileptonic $B \to T\ell^+\ell^-$, $B \to T(\to PP)\ell^+\ell^-$, and $B \to T(\to PV)\ell^+\ell^-$ decays with $\ell=e,\mu,\tau$ based on flavor SU(3) analysis in the standard model ($T$ denotes the light tensor meson, $P$ denotes the light pseudoscalar meson, and $V$ denotes the light vector meson). The hadronic amplitudes of the $B \to T\ell^+\ell^-$ decays are related by the nonperturbative parameters, and all branching ratios of the $B \to T\ell^+\ell^-$ decays are obtained by the experimental data of the branching ratio of $B^0_s\to f^{\prime}_{2}(1525)\mu^+\mu^-$ in three cases, and then the branching ratios of the $B \to T(\to PP)\ell^+\ell^-$ and $B \to T(\to PV)\ell^+\ell^-$ decays are predicted by the narrow width approximation and further considering finite width effects of the intermediate resonances. Compared with the narrow width results, the finite width effects slightly reduce the branching fractions for most decays. However, sizeable finite width effects are found in some near threshold modes. For the subthreshold $K_2^*(1430)\to K\eta'$ relevant channels, the finite width of the tensor resonance can open a nonzero contribution. Compared with the measured $B \to PP\ell^+\ell^-$ and $B \to PV\ell^+\ell^-$ decays, we find that the branching ratios with the tensor resonance states are small. Therefore, other resonances, for example, the vector mesons, the scalar mesons, the axial-vector mesons or their excited states, might give the dominant contributions to the relevant decays. Our results might be tested in current and future experiments.

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Works this paper leans on

66 extracted references · 58 linked inside Pith

  1. [1]

    Navas et al

    S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024) and 2025 update

    2024

  2. [2]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Lett. B 743, 46-55 (2015) [arXiv:1412.6433 [hep-ex]]

    Pith/arXiv arXiv 2015

  3. [3]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 11 (2016), 047 [erratum: JHEP 04 (2017), 142] [arXiv:1606.04731 [hep-ex]]

    Pith/arXiv arXiv 2016

  4. [4]

    Aaij et al

    R. Aaij et al. [LHCb], Phys. Rev. Lett. 125 (2020) no.1, 011802 [arXiv:2003.04831 [hep-ex]]

    arXiv 2020

  5. [5]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 02 (2016), 104 doi:10.1007/JHEP02(2016)104 [arXiv:1512.04442 [hep-ex]]

    Pith/arXiv arXiv 2016

  6. [6]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 08 (2013), 131 doi:10.1007/JHEP08(2013)131 [arXiv:1304.6325 [hep-ex]]

    Pith/arXiv arXiv 2013

  7. [7]

    Aaij et al

    R. Aaij et al. [LHCb], JHEP 12 (2016), 065 doi:10.1007/JHEP12(2016)065 [arXiv:1609.04736 [hep-ex]]

    Pith/arXiv arXiv 2016

  8. [8]

    Isgur, D

    N. Isgur, D. Scora, B. Grinstein and M. B. Wise, Phys. Rev. D 39, 799-818 (1989)

    1989

  9. [9]

    Scora and N

    D. Scora and N. Isgur, Phys. Rev. D 52, 2783-2812 (1995) [arXiv:hep-ph/9503486 [hep-ph]]

    Pith/arXiv arXiv 1995

  10. [10]

    Sharma and R

    N. Sharma and R. C. Verma, Phys. Rev. D 82, 094014 (2010) [arXiv:1004.1928 [hep-ph]]

    Pith/arXiv arXiv 2010

  11. [11]

    Wang, Phys

    W. Wang, Phys. Rev. D 83, 014008 (2011) [arXiv:1008.5326 [hep-ph]]

    Pith/arXiv arXiv 2011

  12. [12]

    T. M. Aliev and Y. Sarac, Phys. Rev. D 111, no.5, 056028 (2025) [arXiv:2411.15952 [hep-ph]]

    Pith/arXiv arXiv 2025

  13. [13]

    K. C. Yang, Phys. Lett. B 695, 444-448 (2011) [arXiv:1010.2944 [hep-ph]]

    Pith/arXiv arXiv 2011

  14. [14]

    T. M. Aliev, H. Dag, A. Kokulu and A. Ozpineci, Phys. Rev. D 100, no.9, 094005 (2019) [arXiv:1908.00847 [hep-ph]]

    Pith/arXiv arXiv 2019

  15. [15]

    Y. B. Zuo, C. X. Yue, B. Yu, Y. H. Kou, Y. Chen and W. Ling, Eur. Phys. J. C 81 (2021) no.1, 30

    2021

  16. [16]

    Khosravi and S

    R. Khosravi and S. Sadeghi, Adv. High Energy Phys. 2016 (2016), 2352041 [arXiv:1503.02883 [hep-ph]]

    Pith/arXiv arXiv 2016

  17. [17]

    Charles, A

    J. Charles, A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Phys. Rev. D 60, 014001 (1999) [arXiv:hep-ph/9812358 [hep-ph]]

    Pith/arXiv arXiv 1999

  18. [18]

    Descotes-Genon, A

    S. Descotes-Genon, A. Khodjamirian, J. Virto and K. K. Vos, JHEP 06 (2023), 034 [arXiv:2304.02973 [hep-ph]]

    Pith/arXiv arXiv 2023

  19. [19]

    X. G. He, Eur. Phys. J. C 9, 443 (1999) [hep-ph/9810397]

    Pith/arXiv arXiv 1999

  20. [20]

    X. G. He, Y. K. Hsiao, J. Q. Shi, Y. L. Wu and Y. F. Zhou, Phys. Rev. D 64, 034002 (2001) [hep-ph/0011337]. 22

    Pith/arXiv arXiv 2001

  21. [21]

    H. K. Fu, X. G. He and Y. K. Hsiao, Phys. Rev. D 69, 074002 (2004) [hep-ph/0304242]

    Pith/arXiv arXiv 2004

  22. [22]

    Y. K. Hsiao, C. F. Chang and X. G. He, Phys. Rev. D 93, no. 11, 114002 (2016) [arXiv:1512.09223 [hep-ph]]

    Pith/arXiv arXiv 2016

  23. [23]

    X. G. He and G. N. Li, Phys. Lett. B 750, 82 (2015) [arXiv:1501.00646 [hep-ph]]

    Pith/arXiv arXiv 2015

  24. [24]

    Gronau, O

    M. Gronau, O. F. Hernandez, D. London and J. L. Rosner, Phys. Rev. D 50, 4529 (1994) [hep-ph/9404283]

    Pith/arXiv arXiv 1994

  25. [25]

    Gronau, O

    M. Gronau, O. F. Hernandez, D. London and J. L. Rosner, Phys. Rev. D 52, 6356 (1995) [hep-ph/9504326]

    Pith/arXiv arXiv 1995

  26. [26]

    S. H. Zhou, Q. A. Zhang, W. R. Lyu and C. D. L, Eur. Phys. J. C 77, no. 2, 125 (2017) [arXiv:1608.02819 [hep-ph]]

    Pith/arXiv arXiv 2017

  27. [27]

    H. Y. Cheng, C. W. Chiang and A. L. Kuo, Phys. Rev. D 91, no. 1, 014011 (2015) [arXiv:1409.5026 [hep-ph]]

    Pith/arXiv arXiv 2015

  28. [28]

    M. He, X. G. He and G. N. Li, Phys. Rev. D 92, no. 3, 036010 (2015) [arXiv:1507.07990 [hep-ph]]

    Pith/arXiv arXiv 2015

  29. [29]

    N. G. Deshpande and X. G. He, Phys. Rev. Lett. 75, 1703 (1995) [hep-ph/9412393]

    Pith/arXiv arXiv 1995

  30. [30]

    Shivashankara, W

    S. Shivashankara, W. Wu and A. Datta, Phys. Rev. D 91, 115003 (2015) [arXiv:1502.07230 [hep-ph]]

    Pith/arXiv arXiv 2015

  31. [31]

    R. M. Wang, X. D. Cheng, Y. Y. Fan, J. L. Zhang and Y. G. Xu, J. Phys. G 48 (2021), 085001 [arXiv:2008.06624 [hep-ph]]

    Pith/arXiv arXiv 2021

  32. [32]

    Y. Qiao, J. H. Sheng, Y. G. Xu and R. M. Wang, Phys. Rev. D 112 (2025) no.3, 033002 [arXiv:2507.17537 [hep-ph]]

    Pith/arXiv arXiv 2025

  33. [33]

    R. M. Wang, Y. G. Xu, C. Hua and X. D. Cheng, Phys. Rev. D 103 (2021) no.1, 013007 [arXiv:2101.02421 [hep-ph]]

    Pith/arXiv arXiv 2021

  34. [34]

    D. Das, B. Kindra, G. Kumar and N. Mahajan, Phys. Rev. D 99 (2019) no.9, 093012 [arXiv:1812.11803 [hep-ph]]

    Pith/arXiv arXiv 2019

  35. [35]

    Ahmed, M

    I. Ahmed, M. J. Aslam, M. Junaid and S. Shafaq, JHEP 02 (2012), 045

    2012

  36. [36]

    Junaid, M

    M. Junaid, M. J. Aslam and I. Ahmed, Int. J. Mod. Phys. A 27, 1250149 (2012) [arXiv:1103.3934 [hep-ph]]

    Pith/arXiv arXiv 2012

  37. [37]

    R. H. Li, C. D. Lu and W. Wang, Phys. Rev. D 83, 034034 (2011) [arXiv:1012.2129 [hep-ph]]

    Pith/arXiv arXiv 2011

  38. [38]

    S. R. Choudhury, A. S. Cornell and N. Gaur, Phys. Rev. D 81 (2010), 094018 [arXiv:0911.4783 [hep-ph]]

    Pith/arXiv arXiv 2010

  39. [39]

    Katirci and K

    N. Katirci and K. Azizi, JHEP 07 (2011), 043 [arXiv:1105.3636 [hep-ph]]

    Pith/arXiv arXiv 2011

  40. [40]

    T. M. Aliev and M. Savci, Phys. Rev. D 85 (2012), 015007 [arXiv:1109.2738 [hep-ph]]

    Pith/arXiv arXiv 2012

  41. [41]

    Vardani and R

    J. Vardani and R. Sain, Eur. Phys. J. C 84 (2024) no.11, 1150 [arXiv:2407.15577 [hep-ph]]

    arXiv 2024

  42. [42]

    Capdevila, J

    B. Capdevila, J. Matias, M. Novoa-Brunet, M. Patel and M. Smith, Phys. Rev. D112 (2025) no.1, 016007 [arXiv:2504.00949 [hep-ph]]

    Pith/arXiv arXiv 2025

  43. [43]

    C. D. Lu and W. Wang, Phys. Rev. D 85, 034014 (2012) [arXiv:1111.1513 [hep-ph]]

    Pith/arXiv arXiv 2012

  44. [44]

    Gratrex, M

    J. Gratrex, M. Hopfer and R. Zwicky, Phys. Rev. D 93, no.5, 054008 (2016) [arXiv:1506.03970 [hep-ph]]

    Pith/arXiv arXiv 2016

  45. [45]

    C. S. Kim and T. Yoshikawa, [arXiv:0711.3880 [hep-ph]]

    Pith/arXiv arXiv

  46. [46]

    Rajeev, N

    N. Rajeev, N. Sahoo and R. Dutta, Phys. Rev. D 103, no.9, 095007 (2021) [arXiv:2009.06213 [hep-ph]]

    Pith/arXiv arXiv 2021

  47. [47]

    Alguer´ o, P

    M. Alguer´ o, P. A. Cartelle, A. M. Marshall, P. Masjuan, J. Matias, M. A. McCann, M. Patel, K. A. Petridis and M. Smith, JHEP 12 (2021), 085 [arXiv:2107.05301 [hep-ph]]

    Pith/arXiv arXiv 2021

  48. [48]

    D. Das, G. Hiller, M. Jung and A. Shires, JHEP 09, 109 (2014) [arXiv:1406.6681 [hep-ph]]

    Pith/arXiv arXiv 2014

  49. [49]

    Buchalla, A

    G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125-1144 (1996) [arXiv:hep-ph/9512380 [hep-ph]]

    Pith/arXiv arXiv 1996

  50. [50]

    Hatanaka and K

    H. Hatanaka and K. C. Yang, Eur. Phys. J. C 67, 149-162 (2010) [arXiv:0907.1496 [hep-ph]]

    Pith/arXiv arXiv 2010

  51. [51]

    Hatanaka and K

    H. Hatanaka and K. C. Yang, Phys. Rev. D 79 (2009), 114008 [arXiv:0903.1917 [hep-ph]]

    Pith/arXiv arXiv 2009

  52. [52]

    M. K. Mohapatra and A. Giri, Phys. Rev. D 104, no.9, 095012 (2021) [arXiv:2109.12382 [hep-ph]]

    Pith/arXiv arXiv 2021

  53. [53]

    Descotes-Genon, J

    S. Descotes-Genon, J. Matias, M. Ramon and J. Virto, JHEP 01, 048 (2013) [arXiv:1207.2753 [hep-ph]]

    Pith/arXiv arXiv 2013

  54. [54]

    Descotes-Genon, T

    S. Descotes-Genon, T. Hurth, J. Matias and J. Virto, JHEP 05, 137 (2013) [arXiv:1303.5794 [hep-ph]]

    Pith/arXiv arXiv 2013

  55. [55]

    Ecker and C

    G. Ecker and C. Zauner, Eur. Phys. J. C 52 (2007), 315-323 [arXiv:0705.0624 [hep-ph]]

    Pith/arXiv arXiv 2007

  56. [56]

    C. Chen, N. Q. Cheng, L. W. Yan, C. G. Duan and Z. H. Guo, Phys. Rev. D 108 (2023) no.1, 014002 [arXiv:2302.11316 [hep-ph]]

    Pith/arXiv arXiv 2023

  57. [57]

    H. Y. Cheng and K. C. Yang, Phys. Rev. D 83 (2011), 034001 [arXiv:1010.3309 [hep-ph]]

    Pith/arXiv arXiv 2011

  58. [58]

    X. G. He, Y. J. Shi and W. Wang, Eur. Phys. J. C 80 (2020) no.5, 359 [arXiv:1811.03480 [hep-ph]]

    Pith/arXiv arXiv 2020

  59. [59]

    X. G. He, G. N. Li and D. Xu, Phys. Rev. D 91, no.1, 014029 (2015) [arXiv:1410.0476 [hep-ph]]

    Pith/arXiv arXiv 2015

  60. [60]

    Yi Qiao, Jin-Huan Sheng, Yuan-Guo Xu, Ru-Min Wang, Phys. Rev. D 112, 033002 (2025)

    2025

  61. [61]

    Y. Qiao, Y. X. Liu, Y. G. Xu and R. M. Wang, Eur. Phys. J. C 84, no.10, 1110 (2024) [arXiv:2404.03857 [hep-ph]]

    Pith/arXiv arXiv 2024

  62. [62]

    Yuan-Guo Xu and Ru-Min Wang, B → P P ℓ+νℓ, in preparation. 23

  63. [63]

    Blake, U

    T. Blake, U. Egede and A. Shires, JHEP 03 (2013), 027 [arXiv:1210.5279 [hep-ph]]

    Pith/arXiv arXiv 2013

  64. [64]

    Aaij et al

    R. Aaij et al. [LHCb], [arXiv:2510.13716 [hep-ex]]

    Pith/arXiv arXiv

  65. [65]

    H. Y. Cheng, C. Y. Cheung, W. Dimm, G. L. Lin, Y. C. Lin, T. M. Yan and H. L. Yu, Phys. Rev. D 48, 3204-3220 (1993) [arXiv:hep-ph/9305340 [hep-ph]]

    Pith/arXiv arXiv 1993

  66. [66]

    S. Y. Tsai and Y. K. Hsiao, [arXiv:2107.03634 [hep-ph]]

    Pith/arXiv arXiv